Mix the patty ingredients, and then gently form four patties of equal size, each about 1 inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of the patties to prevent them from forming a dome as they cook. Refrigerate the patties until ready to grill.
In a skillet over medium heat, fry the bacon until crisp, 10 to 12 minutes, turning occasionally. Transfer the bacon to paper towels to drain. Pour off and discard all
but 2 tablespoons of the bacon fat from the skillet.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness (160°F), 9 to 11 minutes, turning once. During the last 30 seconds to 1 minute of grilling time, place a slice of cheese on each patty to melt, and toast the rolls, cut side down, over direct heat.
Return the skillet over medium heat and warm the bacon fat. Crack the eggs into the skillet and cook until the yolks are still slightly runny, about 3 minutes, turning once. Season with salt and pepper.
Build each burger on a roll with two tomato slices, a patty, two slices of bacon, and an egg. Serve warm.